0.2 Remarks

1.

A standard example of a unitary space is
ℂn with inner product

⟨u,v⟩=∑i=1nui⁢vi¯,u,v∈ℂn.

(2)

2.

Unitary transformations and unitary matrices
are closely related. On the one hand, a unitary matrix defines a
unitary transformation of ℂn relative to the inner product
(2). On the other hand, the representing matrix of a
unitary transformation relative to an orthonormal basis is, in fact, a
unitary matrix.

3.

A unitary transformation is an automorphism. This follows from
the fact that a unitary transformation Tpreserves the
inner-product norm:

Thus, the kernel of T is trivial, and therefore it is an
automorphism.

4.

Moreover, relation (3) can be taken as the definition
of a unitary transformation. Indeed, using the polarization
identity it is possible to show that if
T preserves the norm, then (1) must hold as well.

5.

A simple example of a unitary matrix is the change of
coordinates matrix between two orthonormal bases. Indeed, let
u1,…,un and v1,…,vn be two orthonormal bases, and
let A=(Aji) be the corresponding change of basis matrix
defined by